[Exercise 5.15 in [1]}]
Let \(X_t\)  be a real valued random variable in the filtration of \(\mathcal {F}_t = (X_1,\dots X_{t-1})\)  such that \(\mathbb {E}[X_t\,|\,\mathcal {F}_t] = 0\)  and assume \(\eta >0\) , \(\mathbb {E}[X_t\,|\,\mathcal {F}_t]<\eta ^{-1}\)  a.s. Then with probability at least \(1-\delta \)  for all \(0<t\le T\) ,
\(\sum _{s=1}^tX_s \le \eta \sum _{s=1}^t\mathbb {E}[X_s^2\,|\,\mathcal {F}_s]+\eta ^{-1}\ln (\delta ^{-1})\,.\) 
